MLE for kidney infection data.

We propose the Fréchet–Power Function (FPF) distribution, a novel two-parameter model that combines the bounded support of the Power Function distribution with the heavy-tailed flexibility of a Fréchet-type generator. This combination enables the FPF to capture complex features such as skewness, heavy tails, and diverse hazard rate shapes—limitations present in existing bounded lifetime models. We provide explicit forms for the probability density, cumulative distribution, and quantile functions, along with detailed statistical properties including moments and hazard rate behavior. Parameters are estimated using maximum likelihood, with bootstrap and simulation techniques employed to assess estimator performance. Empirical applications to survival, reliability, and environmental datasets show that the FPF distribution consistently outperforms traditional models in terms of goodness-of-fit and flexibility. This work introduces a powerful and versatile tool for modeling bounded lifetime data, offering enhanced accuracy and interpretability across disciplines.